This relation would be more simple in some cases. For example, in de Sitter space suppose the theory in which \(\mathcal {H}^{(2)}\)  is proportional to \(a^n\)  with \(n > 0\)  for scale factor [1]}. If the coupling between system and environment is weak enough such that the density matrix remains close to Gaussian, then the real part of \(F\)  doesn't grow at late time. In other words, when \(\tau ^{^{\prime }}\)  is close to \(\tau \) , the exponent part in Eq. (REF ) is negligible. So for late time we have just imaginary part of this as following
\(\lim _{\tau \rightarrow 0}Im F_{k k^{^{\prime }} p}(\tau ) = \int _{\tau _0}^{\tau } \frac{d\tau ^{^{\prime }}}{H\tau ^{^{\prime }}} \mathcal {H}^{(2)}_{k k^{^{\prime }} p}(\tau ^{^{\prime }}).\) 
